Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs    [PDF]

Joe P. Chen, Baris Evren Ugurcan
Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. Menger sponge), that is, a centered Gaussian field whose covariance is the Green's function for simple random walk on the graph. Moreover assume that a "hard wall" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on \$\mathbb{Z}^d\$, \$d \geq 3\$, to the fractal setting. Our proof utilizes the theory of transient regular Dirichlet forms, in conjunction with the relative entropy, coarse graining, and conditioning arguments introduced in the previous literature.
View original: http://arxiv.org/abs/1307.5825